answer：First, I need to find the prime factorization of 544 to understand its divisors.544 can be divided by 2 repeatedly:544 ÷ 2 = 272272 ÷ 2 = 136136 ÷ 2 = 6868 ÷ 2 = 3434 ÷ 2 = 17So, 544 = 2⁵ × 17¹.Next, I'll identify the divisors of 544 that are perfect squares. A perfect square has even exponents in its prime factorization.For the prime factor 2, the exponents in the divisors can be 0, 2, or 4 to form perfect squares:- 2⁰ = 1- 2² = 4- 2⁴ = 16For the prime factor 17, since its exponent is 1, it cannot contribute to a perfect square divisor.Now, I'll list the perfect square divisors:1, 4, and 16.Finally, I'll sum these perfect square divisors:1 + 4 + 16 = 21.

answer：Okay, so I've got this problem about an arithmetic sequence. Let me try to unpack it step by step. First, the problem gives me two equations involving the fifth and eleventh terms of the sequence. The equations are:1. ((a_5 - 1)^{2015} + 2016a_5 + (a_5 - 1)^{2017} = 2008)2. ((a_{11} - 1)^{2015} + 2016a_{11} + (a_{11} - 1)^{2017} = 2024)And I need to figure out which statement about (S_{15}) and the common difference (d) is true. The options are:A: (S_{15} = 22), (d < 0)B: (S_{15} = 22), (d > 0)C: (S_{15} = 15), (d < 0)D: (S_{15} = 15), (d > 0)Alright, so let's start by recalling some properties of arithmetic sequences. In an arithmetic sequence, each term is the previous term plus a common difference (d). So, the nth term can be written as:(a_n = a_1 + (n - 1)d)Where (a_1) is the first term.The sum of the first (n) terms, (S_n), is given by:(S_n = frac{n}{2}(2a_1 + (n - 1)d))Alternatively, since (a_n = a_1 + (n - 1)d), we can also write:(S_n = frac{n}{2}(a_1 + a_n))So, for (S_{15}), it would be:(S_{15} = frac{15}{2}(a_1 + a_{15}))But since (a_{15} = a_1 + 14d), this simplifies to:(S_{15} = frac{15}{2}(2a_1 + 14d) = frac{15}{2}(2(a_1 + 7d)) = 15(a_1 + 7d))Hmm, interesting. So, (S_{15}) is 15 times the eighth term, because (a_8 = a_1 + 7d). That might come in handy later.Now, back to the given equations. They both involve (a_5) and (a_{11}), so maybe I can express these in terms of (a_1) and (d).Let's compute (a_5) and (a_{11}):(a_5 = a_1 + 4d)(a_{11} = a_1 + 10d)So, both equations involve expressions of (a_1 + 4d) and (a_1 + 10d). Let me denote (x = a_5 - 1) and (y = a_{11} - 1). Then, the equations become:1. (x^{2015} + 2016(a_5) + x^{2017} = 2008)2. (y^{2015} + 2016(a_{11}) + y^{2017} = 2024)Wait, but (a_5 = x + 1) and (a_{11} = y + 1). So, substituting back, we have:1. (x^{2015} + 2016(x + 1) + x^{2017} = 2008)2. (y^{2015} + 2016(y + 1) + y^{2017} = 2024)Let me simplify these equations:For the first equation:(x^{2015} + 2016x + 2016 + x^{2017} = 2008)Subtract 2008 from both sides:(x^{2015} + 2016x + x^{2017} + 2016 - 2008 = 0)Simplify the constants:2016 - 2008 = 8, so:(x^{2015} + 2016x + x^{2017} + 8 = 0)Wait, that seems a bit messy. Maybe I made a miscalculation. Let me check again.Wait, the original equation is:(x^{2015} + 2016a_5 + x^{2017} = 2008)But (a_5 = x + 1), so:(x^{2015} + 2016(x + 1) + x^{2017} = 2008)Which is:(x^{2015} + 2016x + 2016 + x^{2017} = 2008)Then, moving 2008 to the left:(x^{2015} + 2016x + x^{2017} + 2016 - 2008 = 0)Which is:(x^{2015} + 2016x + x^{2017} + 8 = 0)Wait, that doesn't seem right because 2016 - 2008 is 8, so it's positive 8. So the equation becomes:(x^{2015} + 2016x + x^{2017} + 8 = 0)Similarly, for the second equation:(y^{2015} + 2016(y + 1) + y^{2017} = 2024)Which is:(y^{2015} + 2016y + 2016 + y^{2017} = 2024)Subtract 2024:(y^{2015} + 2016y + y^{2017} + 2016 - 2024 = 0)Which is:(y^{2015} + 2016y + y^{2017} - 8 = 0)So, now we have two equations:1. (x^{2015} + 2016x + x^{2017} + 8 = 0)2. (y^{2015} + 2016y + y^{2017} - 8 = 0)Hmm, that seems a bit complicated. Maybe I can factor out some terms or find a substitution.Looking at the exponents, both equations have (x^{2015}) and (x^{2017}). Let's factor out (x^{2015}):1. (x^{2015}(1 + x^2) + 2016x + 8 = 0)2. (y^{2015}(1 + y^2) + 2016y - 8 = 0)Hmm, not sure if that helps. Maybe I can define a function (f(t) = t^{2015} + 2016t + t^{2017}). Then, the first equation becomes:(f(x) + 8 = 0) => (f(x) = -8)And the second equation becomes:(f(y) - 8 = 0) => (f(y) = 8)So, (f(x) = -8) and (f(y) = 8). Interesting.Now, let's analyze the function (f(t) = t^{2015} + 2016t + t^{2017}).First, notice that 2015 and 2017 are both odd exponents. So, (t^{2015}) and (t^{2017}) are both odd functions. Also, 2016t is an odd function. Therefore, the entire function (f(t)) is an odd function because it's the sum of odd functions.That means (f(-t) = -f(t)). So, if (f(x) = -8), then (f(-x) = 8). Similarly, if (f(y) = 8), then (f(-y) = -8).Looking back at our equations, (f(x) = -8) and (f(y) = 8). So, if (f(x) = -8), then (f(-x) = 8). Therefore, (y = -x). Because (f(y) = 8 = f(-x)), so (y = -x).Therefore, (y = -x). So, (a_{11} - 1 = - (a_5 - 1)). Let's write that out:(a_{11} - 1 = - (a_5 - 1))Simplify:(a_{11} - 1 = -a_5 + 1)Bring the 1 to the other side:(a_{11} = -a_5 + 2)So, (a_{11} + a_5 = 2). That's a useful relation.Now, in an arithmetic sequence, the nth term is (a_n = a_1 + (n - 1)d). So, (a_5 = a_1 + 4d) and (a_{11} = a_1 + 10d). Therefore:(a_5 + a_{11} = (a_1 + 4d) + (a_1 + 10d) = 2a_1 + 14d = 2)So, (2a_1 + 14d = 2), which simplifies to:(a_1 + 7d = 1)Wait, that's interesting because earlier, I noted that (S_{15} = 15(a_1 + 7d)). So, substituting (a_1 + 7d = 1), we get:(S_{15} = 15 times 1 = 15)So, (S_{15} = 15). That narrows it down to options C and D.Now, we need to determine whether (d) is positive or negative. Let's see.We know that (a_5 + a_{11} = 2), and since (a_{11} = a_5 + 6d) (because (a_{11} = a_1 + 10d) and (a_5 = a_1 + 4d), so (a_{11} = a_5 + 6d)), we can write:(a_5 + (a_5 + 6d) = 2)Which simplifies to:(2a_5 + 6d = 2)Divide both sides by 2:(a_5 + 3d = 1)But from earlier, (a_1 + 7d = 1). So, we have:1. (a_5 + 3d = 1)2. (a_1 + 7d = 1)But (a_5 = a_1 + 4d), so substituting into the first equation:((a_1 + 4d) + 3d = 1)Which is:(a_1 + 7d = 1)Which is the same as the second equation. So, no new information there.Hmm, perhaps we can use the function (f(t)) to figure out the relationship between (x) and (y), and thus determine the sign of (d).Earlier, we established that (y = -x), so (a_{11} - 1 = - (a_5 - 1)). Let's write that again:(a_{11} - 1 = - (a_5 - 1))Which simplifies to:(a_{11} = -a_5 + 2)But (a_{11} = a_5 + 6d), so:(a_5 + 6d = -a_5 + 2)Bring (a_5) to the left:(2a_5 + 6d = 2)Divide both sides by 2:(a_5 + 3d = 1)Which is consistent with earlier findings.But how does this help us find (d)?Wait, maybe we can use the function (f(t)) being monotonically increasing. Since (f(t)) is an odd function and the exponents are all odd, the function is increasing because the derivative (f'(t)) is positive for all (t). Let me check that.Compute (f'(t)):(f(t) = t^{2015} + 2016t + t^{2017})So,(f'(t) = 2015t^{2014} + 2016 + 2017t^{2016})Since all the exponents are even, (t^{2014}) and (t^{2016}) are always non-negative, and the coefficients 2015, 2016, 2017 are positive. Therefore, (f'(t) > 0) for all real (t). So, (f(t)) is strictly increasing.Therefore, since (f(x) = -8) and (f(y) = 8), and (f) is strictly increasing, it must be that (x < 0) and (y > 0), because (f(0) = 0 + 0 + 0 = 0). So, (f(x) = -8 < 0) implies (x < 0), and (f(y) = 8 > 0) implies (y > 0).But we also have (y = -x), so if (x < 0), then (y = -x > 0), which is consistent.Now, let's recall that (x = a_5 - 1) and (y = a_{11} - 1). So, (x < 0) implies (a_5 - 1 < 0), so (a_5 < 1). Similarly, (y > 0) implies (a_{11} - 1 > 0), so (a_{11} > 1).Therefore, (a_5 < 1) and (a_{11} > 1). Since (a_{11} = a_5 + 6d), and (a_{11} > a_5), it must be that (d > 0). Because adding a positive multiple of (d) to (a_5) makes it larger.So, (d > 0).Therefore, combining this with (S_{15} = 15), the correct answer is D: (S_{15} = 15), (d > 0).Let me just double-check to make sure I didn't make any mistakes.We started by expressing (a_5) and (a_{11}) in terms of (a_1) and (d), then defined (x) and (y) as (a_5 - 1) and (a_{11} - 1). We transformed the given equations into expressions involving (x) and (y), recognized the function (f(t)) as odd and strictly increasing, leading us to conclude (y = -x). This gave us the relationship (a_{11} + a_5 = 2), which helped us find (S_{15} = 15). Then, by analyzing the signs of (x) and (y), we determined that (d > 0). Everything seems to check out.**Final Answer**boxed{D}

answer：Okay, so I have this problem about an arithmetic-geometric sequence. Hmm, I'm not entirely sure what that is, but I think it's a combination of an arithmetic sequence and a geometric sequence. Maybe each term is both arithmetic and geometric? Wait, that doesn't make sense because arithmetic sequences have a common difference, and geometric sequences have a common ratio. Maybe it's a sequence where each term is the product of an arithmetic term and a geometric term? I'm not too sure, but let me try to figure it out.The problem says the common ratio is q = 3. So, if it's a geometric sequence, each term is multiplied by 3 to get the next term. But it also mentions it's an arithmetic-geometric sequence, so maybe the arithmetic part refers to something else. Maybe the coefficients or something? I'm a bit confused here.Wait, the problem mentions the sum of the first n terms is denoted as S_n. So, maybe it's a standard geometric sequence, and they just called it arithmetic-geometric by mistake? Or maybe it's a different type of sequence where each term is both arithmetic and geometric? That seems complicated. Maybe I should look up what an arithmetic-geometric sequence is, but since I can't right now, I'll try to proceed with what I know.Given that it's a geometric sequence with a common ratio q = 3, I can use the formula for the sum of the first n terms of a geometric sequence. The formula is S_n = a_1 * (1 - q^n) / (1 - q), where a_1 is the first term. So, for S_4, it would be a_1 * (1 - 3^4) / (1 - 3). Let me compute that.First, 3^4 is 81. So, 1 - 81 is -80. Then, 1 - 3 is -2. So, S_4 is a_1 * (-80) / (-2). The negatives cancel out, so it's a_1 * 40. So, S_4 = 40a_1.Now, a_4 is the fourth term of the sequence. Since it's a geometric sequence, each term is multiplied by q each time. So, a_4 = a_1 * q^(4-1) = a_1 * 3^3. 3^3 is 27, so a_4 = 27a_1.The problem asks for S_4 / a_4. So, that would be (40a_1) / (27a_1). The a_1 cancels out, so it's 40/27.Looking at the options, C is 40/27, so that should be the answer. But wait, let me double-check my steps to make sure I didn't make a mistake.First, S_4: 1 - 3^4 is 1 - 81 = -80. 1 - 3 is -2. So, -80 / -2 is 40. So, S_4 = 40a_1. That seems right.a_4: 3^(4-1) is 3^3 = 27. So, a_4 = 27a_1. Correct.Then, S_4 / a_4 = 40a_1 / 27a_1 = 40/27. Yep, that's correct.But wait, the options include 40/9, 80/9, 40/27, and 80/27. So, 40/27 is option C. So, I think that's the right answer.But just to make sure, maybe I misinterpreted the sequence. If it's an arithmetic-geometric sequence, perhaps it's not just a geometric sequence. Maybe it's a combination where each term is the product of an arithmetic term and a geometric term. For example, a_n = (a + (n-1)d) * r^(n-1). So, in that case, the sum S_n would be different.Wait, if that's the case, then the formula for S_n would be more complicated. Let me recall the formula for the sum of an arithmetic-geometric series. It's S_n = (a - (a + (n-1)d)r^n) / (1 - r) + (d r (1 - r^(n-1))) / (1 - r)^2. Hmm, that seems complicated.But in the problem, they just mention it's an arithmetic-geometric sequence with a common ratio of q = 3. Maybe they mean it's a geometric sequence with a common ratio of 3, and the arithmetic part is just the coefficients? Or maybe it's a standard geometric sequence, and the term arithmetic-geometric is a misnomer.Wait, the initial problem statement says it's an arithmetic-geometric sequence with a common ratio of q = 3. So, maybe it's a geometric sequence, and the arithmetic part is not relevant here. Or perhaps it's a different kind of sequence where each term is both arithmetic and geometric, but that would only be possible if the common difference and common ratio are both 1, which isn't the case here.I think the safest assumption is that it's a geometric sequence with q = 3, and the term arithmetic-geometric is either a mistake or refers to something else. Since the problem gives a common ratio, it's likely a geometric sequence.So, going back, S_4 = 40a_1, a_4 = 27a_1, so S_4 / a_4 = 40/27, which is option C.But just to be thorough, let me consider the arithmetic-geometric sequence definition. If a_n = (a + (n-1)d) * r^(n-1), then S_n would be the sum from k=1 to n of (a + (k-1)d) * r^(k-1).In that case, S_n = a * (1 - r^n)/(1 - r) + d * r * (1 - (n) r^(n-1) + (n-1) r^n) / (1 - r)^2.But the problem doesn't give us any information about the arithmetic part, like the common difference d or the first term a. So, I don't think we can use that formula here because we don't have enough information.Therefore, it's more likely that the problem is referring to a geometric sequence with q = 3, and the term arithmetic-geometric is either a misnomer or not relevant to the problem.So, sticking with the geometric sequence approach, S_4 = 40a_1, a_4 = 27a_1, so S_4 / a_4 = 40/27, which is option C.I think that's the correct answer. I don't see any mistakes in my calculations, and the logic seems sound.

answer：First, I need to simplify the given expression ((2m + n)(2m - n) - (2m - n)^2 + 2n(m + n)). I recognize that ((2m + n)(2m - n)) is a difference of squares, which simplifies to (4m^2 - n^2). Next, I expand ((2m - n)^2) using the formula ((a - b)^2 = a^2 - 2ab + b^2), resulting in (4m^2 - 4mn + n^2).Then, I expand (2n(m + n)) by distributing the (2n), which gives (2mn + 2n^2).Now, I substitute these simplified terms back into the original expression:[4m^2 - n^2 - (4m^2 - 4mn + n^2) + 2mn + 2n^2]I distribute the negative sign in front of the second term:[4m^2 - n^2 - 4m^2 + 4mn - n^2 + 2mn + 2n^2]Combining like terms, the (4m^2) and (-4m^2) cancel out, and the ( -n^2 - n^2 + 2n^2) also cancel out, leaving me with:[6mn]Now that the expression is simplified to (6mn), I substitute the given values (m = 2) and (n = -1^{2023}). Since (-1) raised to an odd power remains (-1), (n = -1).Finally, I calculate:[6 times 2 times (-1) = -12]

answer：Okay, so I have this problem where I need to figure out two things among the numbers from 1 to 2015. First, how many of these numbers are multiples of 3. Second, how many numbers leave a remainder of 1 when divided by 3 and a remainder of 2 when divided by 4. Hmm, let me take this step by step.Starting with the first part: finding the number of multiples of 3 between 1 and 2015. I remember that multiples of 3 are numbers like 3, 6, 9, 12, and so on. So, this is an arithmetic sequence where each term increases by 3. The first term is 3, and the last term should be the largest multiple of 3 that is less than or equal to 2015.To find the number of terms in this sequence, I can use the formula for the nth term of an arithmetic sequence: a_n = a_1 + (n - 1)d, where a_n is the nth term, a_1 is the first term, d is the common difference, and n is the number of terms. Here, a_1 is 3, d is 3, and a_n is 2013 because 2013 is the largest multiple of 3 less than 2015 (since 2015 divided by 3 is approximately 671.666, so the integer part is 671, and 671 times 3 is 2013).Plugging these into the formula: 2013 = 3 + (n - 1)*3. Let me solve for n. Subtract 3 from both sides: 2010 = (n - 1)*3. Divide both sides by 3: 670 = n - 1. Add 1 to both sides: n = 671. So, there are 671 multiples of 3 between 1 and 2015.Alright, that seems straightforward. Now, moving on to the second part: finding numbers that leave a remainder of 1 when divided by 3 and a remainder of 2 when divided by 4. This sounds like a problem involving the Chinese Remainder Theorem or solving a system of congruences.Let me write down the conditions mathematically. If a number x leaves a remainder of 1 when divided by 3, that can be written as x ≡ 1 mod 3. Similarly, if it leaves a remainder of 2 when divided by 4, that's x ≡ 2 mod 4. So, I need to find all x in the range 1 to 2015 such that:x ≡ 1 mod 3 x ≡ 2 mod 4I need to solve this system of congruences. Let me see how to approach this. One method is to express x in terms of one modulus and substitute into the other equation.From the first equation, x ≡ 1 mod 3, which means x can be written as x = 3k + 1 for some integer k. Now, substitute this into the second equation: 3k + 1 ≡ 2 mod 4. Let's solve for k.Subtract 1 from both sides: 3k ≡ 1 mod 4. Now, I need to find k such that 3k leaves a remainder of 1 when divided by 4. Let's see, 3 times what gives 1 mod 4. Testing small numbers:3*1 = 3 ≡ 3 mod 4 3*2 = 6 ≡ 2 mod 4 3*3 = 9 ≡ 1 mod 4Ah, so k ≡ 3 mod 4. That means k can be written as k = 4m + 3 for some integer m. Plugging this back into x = 3k + 1:x = 3*(4m + 3) + 1 = 12m + 9 + 1 = 12m + 10.So, x can be expressed as x = 12m + 10. This means the numbers that satisfy both conditions are of the form 12m + 10. Now, I need to find how many such numbers are there between 1 and 2015.Let me find the smallest and largest such numbers in this range. The smallest x is when m = 0: x = 10. The next one is 22, then 34, and so on. The largest x should be less than or equal to 2015. Let me find the maximum m such that 12m + 10 ≤ 2015.Subtract 10 from both sides: 12m ≤ 2005. Divide both sides by 12: m ≤ 2005/12 ≈ 167.083. Since m must be an integer, the maximum m is 167. So, m can take values from 0 to 167, inclusive. That gives us 168 numbers. Wait, hold on, when m=0, x=10, which is within the range, and when m=167, x=12*167 +10=2004 +10=2014, which is also within the range. So, the number of terms is 168.Wait, but earlier when I thought about it, I considered the first term as 10 and the common difference as 12, so the nth term is 12n - 2. Hmm, maybe I made a mistake there. Let me check.If x = 12m + 10, then when m=0, x=10; m=1, x=22; m=2, x=34; and so on. So, the first term is 10, and each subsequent term increases by 12. So, the nth term is 10 + (n-1)*12 = 12n - 2. So, if I set 12n - 2 = 2014, then 12n = 2016, so n=168. So, n=168, meaning there are 168 terms.But wait, in my initial thought process, I thought the number was 167, but now I'm getting 168. Let me verify.If m ranges from 0 to 167, that's 168 numbers because it includes both endpoints. So, the count is 168. But earlier, when I thought about the nth term being 12n - 2, setting that equal to 2014 gives n=168. So, that seems correct.But wait, in the initial problem statement, it's from 1 to 2015. So, 2014 is included, which is correct. So, the number of such integers is 168.Wait, but in the initial analysis, the assistant had 167. Hmm, maybe I need to double-check.Let me list out the numbers:When m=0: x=10 m=1:22 m=2:34 ... m=167:12*167 +10=2004 +10=2014So, starting at m=0, which gives x=10, up to m=167, which gives x=2014. So, the number of terms is 168 because it includes both m=0 and m=167.Wait, but if I use the formula for the number of terms in an arithmetic sequence: n = ((last term - first term)/common difference) +1.Here, first term is 10, last term is 2014, common difference is 12.So, n = ((2014 -10)/12) +1 = (2004/12)+1=167 +1=168. So, yes, 168 terms.So, I think the initial answer of 167 was incorrect, and the correct number is 168.Wait, but in the initial problem, the user wrote:"Integers that leave a remainder of 1 when divided by 3 and a remainder of 2 when divided by 4 form an arithmetic sequence with the first term being 10 and the common difference being 12. Therefore, b_n = 12n - 2, where the largest number that meets the condition is 2014. When b_n = 12n - 2 = 2014, we find n = 167. Thus, among the 2015 positive integers from 1 to 2015, there are 167 integers that leave a remainder of 1 when divided by 3 and a remainder of 2 when divided by 4."Wait, so according to this, n=167, but according to my calculation, it's 168. So, perhaps the assistant made a mistake here.Let me see: if b_n =12n -2, and setting that equal to 2014, 12n -2=2014, so 12n=2016, so n=168. So, n=168, meaning 168 terms. So, the assistant incorrectly concluded n=167, but it should be 168.So, in summary, the number of multiples of 3 is 671, and the number of integers satisfying the two remainder conditions is 168.Wait, but let me double-check the first part as well. The number of multiples of 3 from 1 to 2015.The formula is floor(2015/3)=671.666..., so floor is 671. So, that's correct.For the second part, the number of solutions is 168, not 167.So, perhaps the initial answer had a mistake in the second part.Alternatively, maybe I'm miscounting. Let me think again.If x=12m +10, and m starts at 0, then the number of terms is m=0 to m=167, which is 168 terms. So, 168 numbers.Alternatively, if I think of the sequence starting at 10, with a common difference of 12, the nth term is 10 + (n-1)*12=12n -2. So, setting 12n -2=2014, 12n=2016, n=168. So, n=168, meaning 168 terms.Therefore, the correct number is 168, not 167.So, in conclusion, the number of multiples of 3 is 671, and the number of integers satisfying the two remainder conditions is 168.

answer：Okay, so I need to find the angle between these two planes: First plane: 2x - y + 5z + 16 = 0Second plane: x + 2y + 3z + 8 = 0Hmm, I remember that the angle between two planes is related to the angle between their normal vectors. So, maybe I should start by finding the normal vectors of each plane.For the first plane, the coefficients of x, y, and z are 2, -1, and 5. So, the normal vector for the first plane should be <2, -1, 5>.Similarly, for the second plane, the coefficients are 1, 2, and 3. So, the normal vector is <1, 2, 3>.Alright, now that I have the normal vectors, I think the next step is to find the angle between these two vectors. I recall that the dot product formula can help with this. The formula for the cosine of the angle between two vectors is:cos(theta) = (n1 · n2) / (|n1| |n2|)Where n1 and n2 are the normal vectors, and |n1| and |n2| are their magnitudes.So, let me compute the dot product first.n1 · n2 = (2)(1) + (-1)(2) + (5)(3) = 2 - 2 + 15 = 15Okay, that seems straightforward. Now, I need to find the magnitudes of both normal vectors.For n1: |n1| = sqrt(2^2 + (-1)^2 + 5^2) = sqrt(4 + 1 + 25) = sqrt(30)For n2: |n2| = sqrt(1^2 + 2^2 + 3^2) = sqrt(1 + 4 + 9) = sqrt(14)So, now I can plug these into the cosine formula.cos(theta) = 15 / (sqrt(30) * sqrt(14)) = 15 / sqrt(420)Hmm, sqrt(420) seems a bit messy. Maybe I can simplify that. Let's see, 420 factors into 4 * 105, which is 4 * 15 * 7. So sqrt(420) = sqrt(4 * 105) = 2 * sqrt(105). So, cos(theta) = 15 / (2 * sqrt(105)).Wait, but 15 and 105 have a common factor. 15 is 15, and 105 is 15 * 7. So, maybe I can simplify further.15 / sqrt(420) = 15 / (2 * sqrt(105)) = (15 / sqrt(105)) / 2But 15 / sqrt(105) can be simplified by rationalizing the denominator.15 / sqrt(105) = (15 * sqrt(105)) / 105 = (15 / 105) * sqrt(105) = (1/7) * sqrt(105)So, cos(theta) = (1/7) * sqrt(105) / 2 = sqrt(105) / 14Wait, that doesn't seem right. Let me double-check my steps.Starting from cos(theta) = 15 / sqrt(420). Since 420 = 4 * 105, sqrt(420) = 2 * sqrt(105). So, 15 / (2 * sqrt(105)).Then, to rationalize, multiply numerator and denominator by sqrt(105):(15 * sqrt(105)) / (2 * 105) = (15 / 105) * sqrt(105) / 2 = (1/7) * sqrt(105) / 2 = sqrt(105) / 14Yes, that seems correct. So, cos(theta) = sqrt(105) / 14.Now, to find theta, I need to take the arccosine of sqrt(105)/14.Let me compute that. First, sqrt(105) is approximately 10.24695, and 14 is just 14. So, sqrt(105)/14 ≈ 10.24695 / 14 ≈ 0.7319.Now, arccos(0.7319) is approximately... Let me think. I know that cos(45 degrees) is about 0.7071, and cos(30 degrees) is about 0.8660. So, 0.7319 is between 0.7071 and 0.8660, which means the angle is between 30 and 45 degrees.To get a more precise value, maybe I can use a calculator. But since I don't have one handy, I can estimate it.Alternatively, I can use the inverse cosine function. Let's see, if cos(theta) ≈ 0.7319, then theta ≈ arccos(0.7319).Using a calculator, arccos(0.7319) is approximately 43 degrees.Wait, but earlier I thought it was between 30 and 45, closer to 43. Hmm, okay.But I also remember that sometimes angles between planes can be acute or obtuse, and we usually take the smaller angle. So, if the angle between the normals is 43 degrees, then the angle between the planes is also 43 degrees.But just to make sure, let me recall the formula. The angle between two planes is equal to the angle between their normals, or 180 degrees minus that angle, whichever is smaller. So, in this case, since 43 degrees is less than 90 degrees, that's the angle we want.So, the angle between the two planes is approximately 43 degrees.Wait, but earlier I had an exact expression: arccos(sqrt(105)/14). Maybe I can express it in terms of inverse cosine, but the problem probably expects a numerical value.Alternatively, I can write it as arccos(sqrt(105)/14), but I think the question expects a degree measurement.Alternatively, maybe I made a miscalculation earlier. Let me check again.Wait, cos(theta) = 15 / sqrt(420). Let's compute sqrt(420):sqrt(420) = sqrt(4 * 105) = 2 * sqrt(105) ≈ 2 * 10.24695 ≈ 20.4939So, 15 / 20.4939 ≈ 0.7319Yes, that's correct. So, arccos(0.7319) ≈ 43 degrees.Alternatively, to get a more precise value, I can use the Taylor series or some approximation, but that might be too time-consuming.Alternatively, I can use the fact that cos(43 degrees) ≈ 0.7314, which is very close to 0.7319. So, the angle is approximately 43 degrees.Therefore, the angle between the two planes is approximately 43 degrees.Wait, but let me check if I did everything correctly. Maybe I made a mistake in calculating the dot product or the magnitudes.Dot product: 2*1 + (-1)*2 + 5*3 = 2 - 2 + 15 = 15. That seems correct.Magnitudes:|n1| = sqrt(2^2 + (-1)^2 + 5^2) = sqrt(4 + 1 + 25) = sqrt(30) ≈ 5.477|n2| = sqrt(1^2 + 2^2 + 3^2) = sqrt(1 + 4 + 9) = sqrt(14) ≈ 3.7417So, |n1| * |n2| ≈ 5.477 * 3.7417 ≈ 20.4939So, cos(theta) = 15 / 20.4939 ≈ 0.7319Yes, that's correct.So, arccos(0.7319) ≈ 43 degrees.Therefore, the angle between the two planes is approximately 43 degrees.Alternatively, if I want to express it more precisely, I can use a calculator to find the exact angle.Using a calculator, arccos(0.7319) ≈ 43.0 degrees.So, the angle is approximately 43 degrees.Wait, but sometimes angles between planes are expressed in degrees and minutes. Maybe I can convert 0.0 degrees to minutes.But since it's 43 degrees and approximately 0 minutes, it's just 43 degrees.Alternatively, maybe I can write it as 43 degrees.But to be precise, let me check the exact value.Using a calculator, arccos(0.7319) is approximately 43.0 degrees.So, the angle between the two planes is approximately 43 degrees.Therefore, the final answer is approximately 43 degrees.